A Statistical Test for Partisan Gerrymandering

Transcription

1 A Statistical Test for Partisan Gerrymandering Preliminary Work. DO NOT CITE! Scott T. Macdonell Abstract Is redistricting the result of partisan gerrymandering or apolitical considerations? I develop a statistical test for partisan gerrymandering and apply it to the U.S. Congressional Districting plan chosen by the Republican legislature in Pennsylvania in First, I formally model the optimization problem faced by a strategic Republican redistricter and characterize the theoretically optimal solution. I then estimate the likelihood a district is represented by a Republican, conditional on district demographics. This estimate allows me to determine the value of the gerrymanderer's objective function under any districting plan. Next, I use a geographic representation of the state to randomly generate a large sample of legally valid plans. Finally, I calculate the estimated value of a strategic Republican redistricter's objective function under each of the sample plans and under the actual plan chosen by Republicans. When controlling for incumbency the formal test shows that the Republicans' plan was a partisan gerrymander. 1 Introduction The United States adds an unusual wrinkle to the standard form of representative democracy; every ten years politicians are required to choose voters. Specifically, I consider the decennial redrawing of U.S. Congressional Districts by state I would like to thank Ryan Callihan, Stephen Donald, Laurent Mathevet, Scott Moser, Maxwell Stinchcombe and Thomas Wiseman as well numerous participants in various University of Texas at Austin writing seminars for their helpful comments and insight. Any remaining errors are mine and mine alone. The University of Texas at Austin; 1 University Station, C3100; Austin, TX 78712; smacdonell@utexas.edu; web: 1

2 1 Introduction 2 Tab. 1: 5 District State with 440 Democrats and 560 Republicans District Democrats Republicans (a) Democratic Plan District Democrats Republicans (b) Republican Plan legislatures. This process is designed to avoid an anti-majoritarian problem: Given varying growth rates across the U.S., if Congressional Districts were not occasionally redrawn, some Congresspeople could represent districts of a few thousand individuals, with others representing districts of a few million. Reynolds v. Sims (1964) the U.S. Supreme Court decided that the constitution requires occasional redistricting to ensure that one man's vote in a congressional election is to be worth as much as another's. However, the constitution makes no mention of who should draw new maps; this process has generally been left to state legislatures. This fact gives incumbent politicians substantial power to allocate seats in Congress according to their own interests, possibly against the will of the majority. Their only universal legal constraints are that all districts must be contiguous and (approximately) equipopulous. 1 In As a quick example of the power of gerrymandering, consider a ve district state with 440 voters who always vote for Democrats and 560 voters who always vote for Republicans. Ignoring contiguity, two possible plans are shown in Table 1. The rst plan has one district of 0 Democrats and 200 Republicans, and the other four districts have 110 Democrats and 90 Republicans. In this case Democrats would win an 80% super-majority of the seats while making up a minority of the population. On the other hand, the second plan may initially seem fair as it is proportional; Each district has 88 Democrats and 112 Republicans. However, in this case Democrats from the state would have no representation despite making up nearly half the population. As evidenced by this simple example, the choice of districts can signicantly impact the allocation of seats in Congress. After the 2000 Census, both houses of the Pennsylvania state legislature and 1 Even the contiguity constraint must sometimes be relaxed if, for example, the state has some small islands. Also, dierent states may allow dierent margins of error on the equipopulous constraint.

3 1 Introduction 3 the Governor's mansion were controlled by Republicans. This presented them with the opportunity to draw Pennsylvania's 19 Congressional Districts so as to increase the number of Republicans in the U.S. House of Representatives. In fact, in the 2002 Congressional elections, Republicans won 12 of the state's 19 seats, despite the fact that Pennsylvania seemed to be a swing state that leaned Democratic. 2 This lead several Pennsylvania Democrats to claim their rights to equal representation had been violated and mount a legal challenge to Pennsylvania's districting scheme. This culminated in their case, Vieth v. Jubelirer (2004), reaching the U.S. Supreme Court. The court declined to intervene, deciding that partisan gerrymandering cases were non-justiciable. In Justice Kennedy's controlling opinion he noted that there was no test to appropriately determine if a districting scheme was an unconstitutional attempt to deny members of one party representation, or one based on other, less invidious considerations. 3 However, he left open the possibility that such a test could be developed in the future and that at such a time it could be appropriate for courts to intervene in partisan gerrymandering cases. 4 The purpose of this paper is to develop a statistical test to evaluate claims of partisan gerrymandering. I start by setting up the theoretical optimization problem implied by a claim that a particular plan is a partisan gerrymander: the party in control chose district demographics in order to maximize the expected number of Representatives from their party. Next, in order to calculate the value of the objective function under particular districting schemes, I estimate the relationship between district demographics and the probability of electing a Representative from a given party. Using precinct and census tract data from Pennsylvania in 2000 and GIS techniques, I then randomly generate a large sample of alternative districting schemes that respect contiguity and population equality; this method allows me to calculate the demographics of each district in each sample plan. The nal step of the test compares the estimated value of the objective 2 Two years earlier, Democratic presidential candidate Al Gore won Pennsylvania by a larger margin than he won the national popular vote (which he won). 3 Of course, the justice phrased it dierently: Because there are yet no agreed upon substantive principles of fairness in districting, we have no basis on which to dene clear, manageable, and politically neutral standards for measuring the particular burden a given partisan classication imposes on representational rights. Suitable standards for measuring this burden, however, are critical to our intervention. (Justice Kennedy's Concurrence from Vieth v. Jubelirer (2004)) 4 This opinion was decisive as four justices wished to strike down the Pennsylvania districting plan as an unconstitutional partisan gerrymander while the other four wished to declare partisan gerrymandering cases non-justiciable essentially in perpetuity.

4 1 Introduction 4 function under the actual districting scheme and under the sample schemes in order to test a no partisan gerrymandering null hypothesis. Formally, my null hypothesis is that partisan considerations were not used to develop the districting scheme. I will use plans generated by my algorithm to test this claim in the following manner: If the chances of a disinterested cartographer producing a scheme so favorable to the redistricting party were less than 5%, I then would reject the null hypothesis at the 5% level. This forces acceptance of the alternative, that partisan considerations were used to develop the plan for Pennsylvania. This would imply that charges of partisan gerrymandering are valid by denition. There is a large theoretical and empirical literature on strategic redistricting, though it abstracts away from the geographic nature of the problem. Many papers set up and solve the optimization problem or gerrymandering game faced by the party in charge of redistricting (e.g. Friedman and Holden (2008); Gul and Pesendorfer (2010)) or attempt to nd socially optimal rules which could be imposed on the redistricting process (e.g. Coate and Knight (2007)) Alternatively, many papers take redistricting schemes as given and estimate theoretical quantities like bias 5 in order to determine how much a particular plan favors one party (e.g. Cox and Katz (1999); King and Browning (1987)). Ignoring the geographic nature of redistricting limits the applicability of these theoretical and empirical lines of research. As an example, consider a hypothetical state that leans slightly Democratic overall, but only because of one very Democratic urban area. Perhaps the rest of the state leans slightly Republican. An apolitical cartographer might create a plan with a few districts which include parts of the urban area, but with most districts only covering other parts of the state. Under this plan Democrats would generally win with large majorities in the few urban districts, while Republicans may generally win the rest of the districts more narrowly. This plan might look biased; in this slightly Democratic state the Republicans could expect to win more seats. However, such plans might be quite likely even using non-partisan districting principles. One would expect far dierent outcomes in a more homogeneous state. Clearly, the geographic distribution of voters has important implications for the study of redistricting. To a partisan gerrymanderer, the geographic distribution of voter demographics constrains the set of demographics possible 5 Bias is often dened based on the share of the seats a party could expect to win if they earned half of the statewide vote. If the party should expect to win less than half the seats in this situation, then the plan is biased against them.

5 1 Introduction 5 for Congressional Districts. A state could be thought of as a collection of a large number of small areas (e.g. street addresses, census blocks or tracts, etc...). Redistricting can then be thought of as the process of aggregating those smaller units into a certain number of larger units (Congressional Districts), with the requirement that the larger units be equipopulous and contiguous. Figure 1.1 demonstrates this approach. Figure 1.1a breaks the state of Pennsylvania down into individual census tracts, while Figure 1.1b shows the actual 2002 Congressional District boundaries. An alternative line of research heavily focuses on this geographic interpretation of the problem, but often ignores the important theoretical and empirical results. For instance, papers such as Garnkel and Nemhauser (1970) and Rossiter and Johnston (1981) attempt to identify all possible contiguous and equipopulous solutions to particular redistricting problems, in the hopes of choosing the best. However, this approach is only computationally feasible when the small units are in fact quite large. For example Garnkel and Nemhauser's (1970) method uses counties as their small units and fails for a state with as few as 55 counties. This negates the possibility of using the possibly signicant variation of within county demographic dierences to increase the demographic variation between Congressional Districts. 6 Similar to this paper, Engstrom and Wildgen (1977) and Cirincione, Darling and O'Rourke (2000) take the alternative approach of randomly generating a large number of contiguous and equipopulous redistricting plans in order to evaluate claims that a particular state unconstitutionally used race as a predominant factor during redistricting. This paper provides the rst empirical test of theoretical partisan gerrymandering predictions using a single redistricting plan. 7 Using theoretical and empirical results, I design the test based on a random sample of geographically allowable plans. While scholars have previously used random redistricting methods to test racial gerrymandering claims, they've arbitrarily chosen their test statistics. For instance, Cirincione, Darling and O'Rourke (2000) simply count the number of majority-minority districts in each plan (actual and randomly 6 Increases in computational power provide little hope of redeeming this approach. Altman and McDonald (2011) suggest that even for the modestly sized Wisconsin, when using census blocks, the number of potential districting schemes these methods would have to check may be on the order of the number of quarks in the universe. 7 Others such as Gelman and King (1994) empirically test partisan gerrymandering predictions. However, they require a large number of redistricting plans in their analysis. Courts need the capability to evaluate a claim that an individual plan is a partisan gerrymander.

6 1 Introduction 6 Fig. 1.1: Pennsylvania (a) Census Tracts (b) 2002 Congressional Districts

7 1 Introduction 7 generated). 8 This ignores the possibility that there is a range of population levels where minority representation is likely, yet uncertain. 9 Given these arbitrary test statistics, it may not be surprising that tests based on random samples of possible plans have not yet been used to evaluate claims of partisan gerrymandering. Do we simply count the number of Democratic districts? Is a district Democratic if it contains more registered Democrats? What if there are many independent voters?... I address this issue by formally modeling the optimization problem faced by a partisan redistricter and using the estimated value of their objective function as my test statistic. My approach has the added value of being agnostic towards questions of fairness. For instance, Coate and Knight (2007) and Dopp (2011) try and determine redistricting principles that would induce more optimal districting schemes. Justice Kennedy was quite clear in his controlling opinion in Vieth v. Jubelirer (2004) that choosing among such principles was essentially a political question; any principle would likely favor one party over the other. Here, my test is not based on fairness or optimality, but on how abnormally benecial a plan is to a particular party. Courts could avoid ruling based on whether a plan was fair or unfair. Instead, if the chances of a non-partisan cartographer producing a plan so favorable to the redistricting party were remote, courts could judge the plan a partisan gerrymander. The remainder of this paper proceeds as follows: Section 2 formally denes the partisan gerrymanderer's optimization problem and characterizes the theoretically optimal solution ignoring geography. Section 3.1 discusses some of the data and estimates the parameters of the objective function. Section 3.2 describes the geographic data I use and the algorithm which generates my random sample of alternative plans. Section 3.3 provides the results. Section 4 further discusses the results of my analysis, explores some of the limitations of my approach, and examines further directions for related work. Section 5 concludes. 8 However, they do explore the implications of a few alternative thresholds for when a district should be considered majority-minority. 9 Engstrom and Wildgen (1977) begin to address this concern. Their test statistic assigns a 1 to each district in a plan where less than 45% of the population is a minority, a 3 in each district where minorities make up more than 55% of the population and the appropriate weighted average for districts in between. However, they do not provide a model or empirical estimates to justify this particular choice.

8 2 Theory 8 2 Theory Partisan gerrymandering of U.S. House districts is generally assumed to be done with the aim of increasing the number of Representatives from the redistricter's own party in Congress. A partisan gerrymanderer's task is then to divide a state into N Congressional Districts in such a manner as to maximize the expected number of districts which elect a representative of her party. She may achieve her objectives by choosing the demographics for each district. Without loss of generality, assume that the state is being gerrymandered by Republicans. Let x i be a vector denoting the demographics of interest in district i. Suppose an element of this vector is a scalar Republicaness measure, r i. I will discuss the interpretation and denition of r i further in Section 3.1. For now, assume that the only constraints the redistricters face are that each resident of the state must be in exactly one district, and that the districts must be of equal population. Let x (r) be the value of the demographics (Republicaness) for the whole state. It can be shown that this implies the alternative constraint P N i=1 xi N = x. 10 Of course, geography and correlation across demographics types add additional constraints, but ignore those until Section 3.2. Let y i represent the outcome of the upcoming election in district i. Dene y i 1 if district i elects a Republican, and y i 0 if it elects a Democrat. Assume that there exists some known function, F (x i ) P rob(y it = 1 x i ), which gives the probability of electing a Republican given district demographics. Then the gerrymanderer's optimization problem is as follows: max {x i} N i=1 N F (x i ) s.t. (2.1) i=1 N i=1 x i = x, x x i x, i (2.2) N where x and x represent the lower and upper bounds of the demographics. These could simply reect the fact that it is impossible to make districts that are more than 100% (less than 0%) of a certain demographic type, or one could be 10 So long as we suppose x measures the percentage of voters who are of various demographic types. For example, suppose one of those types is Hispanics. Given a particular plan, if we want to increase the portion of district one that is Hispanic by 1% we must remove 1% of the population from the district (choosing all non-hispanics) and exchange them with Hispanics from other districts (by the population equality constraint). This process would keep the average percent Hispanic constant across the districts.

9 2 Theory 9 more restrictive with the bounds in an attempt to capture some yet unmodeled constraints faced by redistricters, such as contiguity. 11 non-trivial for all demographics of interest (x, x x). Assume the problem is For the purposes of my statistical test, I could end this section with only equation 2.1. All I need is an objective function to evaluate under each districting scheme I consider. However, it is instructive to consider what a solution to this geography-blind optimization problem should look like. Comparing the solution to this version of the gerrymandering problem with the chosen plan provides a useful demonstration of the importance of the yet unmodeled geographic constraints. Assume that r i is the only demographic of interest which can vary across districts (x i is a scalar equal to r i ). 12 Assume that F ( ) is a smooth, symmetric, S shaped distribution. 13 Then, the following theorem applies and will allow me to nd the optimal solution given a specic parameterization of the optimization problem: Theorem 1. For some m {0, 1, 2,..., N 1} one of the following two demographic proles will be a solution to equations 2.1 and 2.2: 1. r i = r i {1,..., m} (if m = 0, then this holds for none of the districts) and r j = q j {m + 1, m + 2,..., N} where q solves (N m)q+mr N = r. 2. r i = r i {1,..., m} and r j = r j {m + 2, m + 3,..., N} and r m+1 = p where p solves mr+(n m 1)r+p N = r. For a proof of this Theorem please see Appendix A. Once the optimization problem has been parameterized, this theorem takes an N dimensional optimization problem and allows one to nd a solution by checking no more than 2N candidate solutions. 14 crack result already common in the literature. 15 Furthermore, this Theorem mirrors the pack and The Republican redistricters will pack many of their opponents into districts where the Democrats will generally win with overwhelming majorities. However, in the the rest of the districts the redistricter will spread the population more evenly (crack) such that 11 It is more appropriate to simply include the geographic constraints as in Section I will justify this assumption in Section Formally, assume z R s.t. y > 0, 1 F (z+y) = F (z y) and that w < z, F (w) > 0 and that F ( ) is continuously dierentiable. 14 Some of the demographic proles that satisfy Theorem 1 may not satisfy the constraints in equation 2.2. Therefore, one also needs to check the feasibility of each candidate solution under this Theorem. 15 For a discussion of this common result as well as an example of the rare paper which nds conicting results see Friedman and Holden (2008)

10 3 Statistical Test 10 the Republicans have more moderate majorities. The logic is that the Republicans are certain to lose a few districts, but with many Democrats excluded from the remaining districts the Republicans are likely to capture a large majority of those. 3 Statistical Test 3.1 Estimation of F ( ) In order to perform my statistical test I need to be able to evaluate equation 2.1 for any potential districting scheme. This will require that I be able to determine {x i } N i=1 under any potential districting scheme, and that I estimate F ( ). First, I discuss the dataset I use in this subsection. Then, I formalize my statistical model and discuss results Data In order to complete the estimation stage of my analysis I used data from a variety of sources. First, I have presidential votes by Congressional District for provided generously by Sean Theriault. This allows me to calculate a useful measure of Republicaness. Specically, I assume that r i for any Congressional District is equal to the Republican presidential candidate's share of the major party vote in the most recent presidential election in district i minus the same share nationwide. 16 Using this measure confers a number of advantages. For one, it directly measures how much more or less Republican a particular area is relative to the rest of the country. This leads to it being consistent across time. Estimating F ( ) using multiple elections cycles and other demographics one might have to worry about a particular group's allegiance to each party changing over time. It seems less likely that there would be a signicant change in how people who prefer Republican presidential candidates feel about Republican Congressional candidates. Furthermore, it inherently controls for the possibility that in the most recent election the Republican candidate may have been especially (un)appealing relative to his opponent. 16 For example, suppose that nationwide in 2008 John McCain received 45% of the votes that were cast for either himself or Barack Obama. Also, suppose that in district i John McCain got 40% of the votes that were cast for either himself or Barack Obama. Then r i = 0.05 for district i in 2008 and 2010.

11 3 Statistical Test 11 Also, I have all U.S. House of Representatives election outcomes for from ICPSR study 6311 (King, 2006). 17 This dataset also contains information on the presence of any incumbents, and their party. 18 Additionally, I have a large number of other demographic variables by Congressional District from coming from a dataset produced by David Lublin. 19 However, as I will discuss along with the estimation, it seems as though a partisan gerrymanderer need not worry about other demographics after controlling for how Republican a district is Statistical Model and Estimation Let ( ) be the logistic function. In order to facilitate estimation, assume that F (x i ) = (α + βx i ). β is a vector of coecients and α is some constant. Now, F ( ) can be estimated according to the standard logit model. Prior to discussing the results of the estimation, I should briey discuss incumbency, a variable conspicuously absent up until now. In general, incumbents nearly always win if they run for reelection. For instance, between 1972 and 1990, there was a Democratic incumbent in 2140 House races. The incumbent won % of these races. 20 However, all incumbents eventually do not run again (retirement, death, scandal...). Furthermore, districting schemes tend to be long lived (10 years). So, there is a strong possibility that any seat will become an open seat under the implemented scheme. Therefore, gerrymanderers may be particularly concerned with how their party would fair in each district were the seat there open. Then, one should think of F ( ) as the probability of electing a Republican, conditional on the seat being open. Since the gerrymanderer may not know who will be retiring, facing a scandal, or dying over the next ten years, they should maximize the sum of the probabilities that they win each seat conditional on that seat being open. After all, the redistricter has far more control over elections to open seats. Alternatively, the gerrymanderer may be particularly concerned with the next election. In this case incumbency plays an important role and it may be 17 This dataset contains data going further back in time. However, I chose not to use it as Wallace's 1968 run for president and concerns about the earlier Dixiecrats could have confounded my Republicaness measure. 18 This data is available for all of the more recent elections in pdf format in CQ Weekly publications, released around mid-april following each congressional election. Future versions of this work will incorporate this more recent data. 19 Currently available at such races featured a Republican incumbent. In these races the incumbent won % of the time.

12 3 Statistical Test 12 Tab. 2: Summary Statistics r i % Black % Hispanic % Urban Median Age Mean Std Dev obs (a) Incumbentless r i % Black % Hispanic % Urban Median Age RepIncumb DemIncumb Mean Std Dev obs (b) With Incumbents appropriate to include variables related to incumbency in the x i s. Therefore, I estimate F ( ) under both assumptions. I base my estimates only on elections in the years for which I currently have data on election outcomes ( ). Also, it may have been possible that some districts were designed specically with some information about the immediately upcoming elections in mind. Thus, in order to remove this possible source of bias, I ignore years ending in I also ignore the few districts which elected independents. This leaves me with 3429 U.S. House races o of which to estimate F ( ). The specications focusing on open seats limit the data to 333 U.S. house races. I also estimate F ( ) including demographic variables other than r i, specically: percent Black, percent Hispanic, percent urban, and median age. Data on the number of Hispanics was missing for some districts. Therefore, the estimates using the additional demographics were based on fewer U.S. House races. I provide summary statistics for my variables in incumbent-less elections during the relevant years in Table 2. The results of my estimation are presented in Table 3. Estimated coecients are presented with standard errors in parentheses beneath each estimated coecient. In the rst two columns I report results for districts without incumbents. In column (1) I present the results based on using r i as the only demographic of interest. Clearly, districts that are more Republican are signicantly more likely to elect a Republican to represent them in Congress. Also, it appears that districts which were about as Republican as the nation as a whole tended to 21 Also, my results from the 1980 presidential election by Congressional District would not have matched up with the Congressional Districts in 1982, due to the intervening redistricting.

13 3 Statistical Test 13 favor Democratic candidates over this time period (the estimated constant was signicantly less than 0). Additionally, I explore the value of controlling for other demographics in column (2) by including the median age of residents as well as the percentage of residents who were Black, Hispanic or Urban. While Black voters tend to choose Democratic candidates, controlling for this demographic does not seem to improve my estimates. In fact, none of the added coecients are signicant at even the 10% level. Performing a Wald test on the restriction that all of the added coecients are zero yields a p-value of This test further suggests that controlling for other demographics adds little to my specication of F ( ). Since I'm already controlling for how Republican a district is, also controlling for another variable (such as % Black) which predicts how Republican a district is does not lead to better predictions of electoral outcomes. Therefore, for the rest of this paper I will assume redistricters ignore other demographic variables. The last two columns report results for the estimation when controlling for incumbency. This was done by including dummy variables for the presence of Republican or Democratic incumbents and two variables interacting r i with those dummies. Column (3) ignores demographics beyond r i and incumbency. Here, the coecients on both dummies have the expected sign and are signicant at the 1% level. The coecients on both of the interaction terms are negative and signicant. The aect of Republicaness on electoral outcomes appears to be muted in the presence of incumbents. Alternatively, I include additional demographics in column (4). Here one of the added coecients is signicant at the 10% level (with a p-value of 0.098). However running a Wald test on the restriction that all of the added coecients are zero yields a p-value of As such for the rest of this paper I will use columns (1) and (3) as my estimates of F ( ). With F ( ) estimated it is now possible to determine the theoretical solution to the optimization problem from equation 2.1 using Theorem 1. Assuming the F ( ) implied by column (1) from Table I dened r and r such that Bush's share of the vote in any district was bounded by [0, 1]. Checking all candidate solutions implies that the theoretically optimal Republican districting scheme for Pennsylvania in the 2000's is one in which six districts were completely Democratic (Gore would have received 100% of the vote) and r i = in 22 This Theorem does not apply to the specication from column (3) as it does not allow for variables like incumbency.

14 3 Statistical Test 14 Tab. 3: Estimation Results (1) (2) (3) (4) r i *** *** *** *** (1.790 ) (2.256 ) (1.790) (2.212) r i RepIncumb *** *** (2.316) (2.682) r i DemIncumb *** *** (2.064) (2.505) %Black (1.754) (0.707) %Hispanic (2.057) (0.809) %U rban * (0.750) (0.324) M edian Age (0.046) (0.021) RepIncumb 2.672*** 2.816*** (0.182) (0.205) DemIncumb *** *** (0.161) (0.185) Constant *** *** (0.133) (1.402) (0.133) (0.655) obs *Signicant at the 10% level **Signicant at the 5% level ***Signicant at the 1% level

15 3 Statistical Test 15 the rest of the districts. (Bush would have received about 70% of the vote in these districts.) The estimates from column (1) imply that the Republicans could have expected to win of the 19 seats under this plan. 3.2 Generation of Sample Plans In order to perform my statistical test I generate a large sample of plans randomly drawn so as to respect contiguity and population equality. I then evaluate each plan according to the objective function developed in Section 2 and my estimates of F ( ) from Section Data I make use of two Census 2000 TIGER/Line shapeles publicly available from Esri. 23 Each shapele represents the state of Pennsylvania broken down by geographic boundaries from the year One splits the state into 9418 precincts, 24 while the other breaks the state into 3135 census tracts. Additionally, I merged the precinct shapele with the 2000 election results and demographic data by precinct available from the Federal Elections Project. 25 (Lublin and Voss, 2001) I also merged the tract shapele with the 2000 census population counts by tract downloaded through the American Fact Finder available on census.gov. Additionally, I downloaded a shapeles representing Pennsylvania broken down by 2000 and 2002 Congressional District from the Census Bureau. 26 Precincts are the smallest level at which votes are counted. This dataset then allows me access to the demographics of interest at the most disaggregated level available to a redistricter. Unfortunately, the algorithm I discuss in Section took too long to divide 9418 precincts into 19 contiguous, equipopulous districts. Therefore, I used the elections data at the precinct level to estimate the number of votes received by Bush and Gore in each census tract. I estimated the number of votes received by a candidate within a tract as the sum of the votes received in each precinct times the percentage of that precinct within the tract Currently all available from 24 referred to as voting districts 25 The merge was generally straightforward, but the two datasets didn't always match up exactly. Please contact me for access to this data as well as a readme le explaining my approach to the merge So, if tract 1 contained 50% of precinct A and 25% of precinct B and no other area, then I estimated the number of votes received by Gore in tract 1 as.5 times the number of votes

16 3 Statistical Test 16 Looking at the two maps, it appeared that a large majority of all precincts lay entirely within a larger tract. Therefore, this method estimated which tract got most precincts' votes with zero error. I also used the same method with the precinct shapele and the 2002 Congressional District shapele to estimate the number of votes Bush and Gore received in each 2002 Congressional District in Pennsylvania. Precincts were also unlikely to lie in multiple Congressional Districts. This estimate should also be very accurate by a similar argument. Additionally, I used the same method to estimate the population of each tract living in each of the 21 Congressional Districts used for the 2000 elections. Generally, a tract was entirely within one district. So, it's population was accurately counted as all living in that district. For those interested in the GIS techniques which led to these estimate please see Appendix B Algorithm The rst part of my algorithm follows directly from Cirincione, Darling and O'Rourke (2000). It starts with the map of Pennsylvania by census tract with no tracts assigned to any Congressional District. First, it randomly chooses an unassigned tract and adds it to District 1. Then it randomly chooses an unassigned tract which neighbors the now growing District 1 and adds it to the district. This last step repeats until the population of the district reaches the ideal district population (646,371), or there are no more unassigned neighbors. 28 Then the above repeats for each of the other 19 districts. (It randomly selects an unassigned tract and assigns it to district 2 and then randomly selects an unassigned neighbor...). If any tracts are left unassigned at the end they are randomly chosen and assigned to a neighboring district. 29 The original Cirincione, Darling and O'Rourke (2000) algorithm required that the population of each district be within 1% of ideal or the algorithm would simply restart, throwing out the plan and starting from scratch. Under this restriction my algorithm always started over. Instead, my algorithm only restarts if at least one district isn't within 50% of the ideal district population. Otherwise, it attempts to x received in precinct A plus.25 times the number of votes received in precinct B. 28 There may be no unassigned neighbors if the construction of an earlier district caused there to be a few unassigned tracts surrounded by already assigned tracts. If the algorithm started creating a district from one of these tracts, it would not be able to make it large enough before running out of unassigned neighbors. 29 It is possible that all of the tracts neighboring an unassigned tract are also unassigned. In this case the algorithm moves on, randomly selecting other unassigned tracts until it nds one with a neighbor assigned to a Congressional District. If the tract chosen has multiple neighbors assigned to dierent districts, the district is chosen randomly from among the possible choices.

17 3 Statistical Test 17 any population inequality by randomly choosing tracts which neighbor another district and reassigning them if the neighbor has a lower population. (If a tract neighbors multiple other districts, one of the neighbors is randomly chosen.) This continues until the population of all Congressional Districts is within 5% of the ideal district size. One can think of plans generated by this algorithm as samples of the plans that might be produced if we tasked an otherwise disinterested cartographer with dividing a state into equipopulous, contiguous districts. She might start picking a small part of the state and decide that should be in one district and then subsequently add small neighboring bits of the state until the district was large enough. Then she might repeat the same process for the other districts she was required to create. If she got stuck, instead of starting from scratch she might choose to reallocate small areas until she achieved an appropriate solution. 3.3 Results My approach generates districting schemes which are apolitical and random, but which also respect the population equality and contiguity constraints required of legal plans. These are exactly the types of alternative plans against which one should compare the actual plan. The purpose of the test is to determine whether partisan considerations guided the design of the redistricting scheme. Again, the null hypothesis is that partisan considerations were not used to develop the plan for Pennsylvania. If the chances of a disinterested cartographer producing a scheme so favorable to the redistricting party were less than 5%, then we should reject the null hypothesis at the 5% level. I would then be forced to accept the alternative, that partisan considerations were used to develop the plan for Pennsylvania. This would imply that charges of partisan gerrymandering are valid by denition. The program to implement my algorithm was coded in R and my code made use of packages designed to deal with this type of geographic problem. 30 Using my program I generated a sample of 10,000 equipopulous, contiguous districting schemes for Pennsylvania. 31 For each sample plan, I used the list of tracts contained within each district and the estimated election results by tract 30 Specically, see Altman and McDonald (2011) 31 This took about 2 days. As of the time I'm writing this, my program is currently rerunning to generate a new sample of 10,000 districting schemes. It seems unlikely that the change will aect my results, but the larger sample will be used in future versions of this work.

18 3 Statistical Test 18 Tab. 4: Results Specication from Table 3 (1) (3) Range of Expected to to Actual Expected Percentile of Actual to calculate the number of votes received by Bush and Gore in each district. 32 From there, the calculation of r i for each district in each sample plan (and the actual plan) was straightforward. I used the estimates of how many people in each tract lived in each of the preredistricting districts similarly to determine the population of each new district that came from each of the old districts. I assumed that incumbents would then run in whichever new district contained the largest number of their former constituents. From here it was straightforward to determine which districts had incumbents from which party. 33 Incumbency dummies were generated for the actual plan by looking at which incumbents ran in which of the new districts. 34 Then, using the estimates of F ( ) from columns (1) and (3) of Table 3, I estimated the value of the partisan redistricter's objective function, Equation 2.1, for each sample plan as well as for the actual 2002 Congressional Districting scheme. The results are reported in Table 4. When ignoring incumbency (Specication (1)), the estimated value of the objective function among the sample plans ranged from to The same value for the actual plan was This was at the approximately the 77th percentile of the value for the sample plans. Therefore, using this specication I cannot reject the null hypothesis that partisan considerations were not used to develop the plan for Pennsylvania. If one thinks redistricters are farsighted and we should not be considering the interaction between redistricting and current incumbents, then my test cannot prove intent. However, it still shows that the chosen plan was particularly favorable for Republicans. These results suggest that only a small fraction ( 23%) of 32 Technically, these numbers are estimates as the number of votes in each tract are estimates. However, as discussed earlier those estimates generally (though not always) should have zero error. 33 Sometimes, two Democratic (Republican) incumbents would have decided to run in the same district. In this case the DemIncumb (RepIncumb) dummy equaled Of the 21 Representatives elected in 2000, there was only one that retired without running in However, most of his constituents were combined with the district of another Democrat (likely prompting the retirement). So, my method would have produced the appropriate incumbency variables under the actual plan as well.

19 4 Discussion 19 possible plans were better for Republicans than the one they chose. The results are much stronger under Specication (3). Republicans could expect to elect more Representatives under their chosen plan than under 96.25% of the sampled plans. We should then reject the non-partisan null-hypothesis at the ve percent level. We must then accept the alternative, that partisan considerations were used to choose the redistricting plan. This implies that the plan is a partisan gerrymander by denition. Bear in mind that this is a working paper and these results should be considered preliminary. 4 Discussion Assuming gerrymanderers are only concerned with the next election, my test conrms that the Pennsylvania plan was in fact a partisan gerrymander. However, if gerrymanderers are concerned with the long run, it may make sense for them to forgo considering the current incumbents while designing their plan. They should then be concerned with how candidates from their party would do in an election to an open seat (which all seats may eventually become). However, my results suggest that the Republicans redistricting Pennsylvania in 2001 were especially concerned with the next election. This seems to match with Pennsylvania's electoral history for the rest of the decade. While in 2002 and 2004, Republicans captured 12 of Pennsylvania's 19 seats, this trend was reversed in the next two elections. Under the same districting scheme in 2006 the Democrats captured 11 seats, and then 12 in The chosen plan worked well in the 2002 election, but eventually Democrats were able to elect a majority of the state's Representatives. This suggests that gerrymanderers are particularly interested in the next election. In Pennsylvania it seems Republicans chose the plan under which they would do the best in the immediately following election. If true, then the appropriate specication is likely the one under which I was able to conrm partisan gerrymandering. Also, in other cases even if my test fails to show intent, it still provides valuable information about eect. If courts were to decide that plans which signicantly favored one party were illegal because they had the eect of a partisan gerrymander they could easily use a less restricted version of my test. An obvious criteria to use might be to require that any plan yield a test statistic within the middle 50% of those of sampled plans. This criteria would not be based on proving intent, but on demonstrating eect. This rule would ban

20 4 Discussion 20 plans under which the redistricting party did better than could be expected, on average, under plans generated by apolitical processes. While this would not eliminate the possibility of partisan gerrymandering, it would add an additional constraint. For instance, the plan for Pennsylvania tested here would fail to meet this stricter criteria under both specications. While my test makes some rather specic assumptions, the method is easily adapted to alternative conceptions of partisan gerrymandering. The theory section could be readily modied to use another objective function. 35 For instance, one might want to include some level of risk aversion, where the redistricter is particularly concerned about how well they do when the opposing party has a strong election year. Empirically, one might want to specify a dierent functional form for F ( ), or include more demographics in the estimation. In fact, earlier in this project I estimated F ( ) using kernel density procedures, avoiding any functional form assumptions at all. However, with very little data o of which to base estimates in the the tails of the distribution of r i, it seemed wise to switch to parametric methods. This may be less of a concern once I update the dataset to include more recent elections. Also, alternative random algorithms could be used to generate the sample of alternative plans. Perhaps one thinks a disinterested cartographer would approach the problem dierently than the algorithm proposed here. 4.1 Limitations Again, all results should be considered preliminary. This paper is a work in progress. Beyond that, several potential criticisms of my approach seem apparent. Obviously, opinions may dier as to the actual objective of a partisan gerrymanderer. I believe I used the most obvious choice: maximizing the expect number of Representatives from the gerrymanderer's party. However, nothing about my test requires this specic objective function, and an alternative could easily be used if preferred. Perhaps the most signicant concern with my approach is that the redistricter could have access to some unobserved information about the population or the upcoming elections. I deal with this as a source of bias for my estimation by only estimating o of elections at least one cycle removed from redistricting. It seems unlikely that in 2001 redistricters had special information about the elections that were not going to occur until So long as that function could be estimated using available data.

21 4 Discussion 21 It is still possible that redistricters had some unobserved information that could have made a particular plan more (or less) attractive than my estimates would suggest. While this is a potential source of error for my test, I argue one should only expect it to increase the incidence of type II errors. If redistricters were not using partisan objectives to choose a plan, there is no reason to suspect that any additional information they had would tend to lead them to choose a plan that looked especially partisan based on my estimates (thus causing a type I error). The alternative is that the redistricter was using partisan objectives to choose a plan. If this was the case and the unobserved information caused an error, it would have to be a type II error (failing to reject non-partisan motives when the redistricter was in fact partisan). Currently the estimation of F ( ) is based on only one demographic and less than current data. In Section I made the case that one should treat deviation from national Republican presidential vote share as a sucient statistic for other demographics that may be of interest to a partisan redistricter. However, there is nothing to prevent one from using more information in the estimation of F ( ). I assumed an incumbent would run in a newly created district if that district contained more of his former constituents than any other district. This may be an inappropriate assumption. Suppose 51% of an incumbents constituents ended up in district 1 and 49% ended up in district 2. My assumption implies that incumbent will run in district 1 during the next election. That implication may be wrong, especially if district 2 would be an otherwise open seat whose demographics favor that incumbent's party. I could remedy this issue by formally modeling the incumbent's choice of which district to run in. I hope to do so in future versions of this works. However, in the case of Pennsylvania, this would be a game with 21 players, each choosing 1 of 19 districts in which to run. Accounting for this feature may unnecessarily complicate the model. Ideally, statistical tests are based on random samples drawn from some population. There is nothing to guarantee that my algorithm generates plans which are randomly drawn from the set of contiguous equipopulous plans. Given the size of the redistricting problem, generating such a random sample seems computationally infeasible. As already discussed, there is no feasible way to generate the entire population of such plans, and taking a random sample would seemingly involve randomly assigning each tract to a Congressional District and then throwing out the plan and starting over if you had not created a contiguous, equipopulous plan. Such an approach would almost always create an invalid